Is every unitary representation a direct sum of irreducible subprepresentations?

Question

I've read that any unitary representation of a compact group decomposes as a Hilbert space direct sum of irreducible representations. In the book I'm reading this is stated as a prong of the Peter-Weyl theorem. It seems to me that this should just be a straighforward application of Zorn's lemma and trans finite induction since the intersection of a nested sequence of closed invariant subspaces is again closed and invariant (edit: it just occurred to me that this intersection may be trivial). This doesn't seem to use compactness of G. Is this right or am I missing something? If such an argument doesn't work, can I derive this fact for compact G from the other parts of the Peter Weyl theorem, namely that $L^2(G)$ is a Hilbert space direct sum of the irreducible unitary representations of G (which are finite dimensional)?